Munich Personal RePEc Archive
The Use of Divisia Monetary Aggregates in Nominal GDP Targeting
Barnett, William A. and Su, Liting
University of Kansas, University of Kansas
25 May 2015
Online at https://mpra.ub.uni-muenchen.de/67040/
MPRA Paper No. 67040, posted 03 Oct 2015 06:10 UTC
The Use of Divisia Monetary Aggregates in Nominal GDP Targeting
by
William A. Barnett
1University of Kansas, Lawrence, Kansas and
Center for Financial Stability, New York City
Liting Su
University of Kansas, Lawrence, Kansas
May 25, 2015
1 Corresponding author: email willliamabarnett@gmail.com. Phone 785-864-2844. Fax: 785-864- 5760.
Abstract: One of the hottest topics in monetary policy research has been the revival of the proposal for βnominal GDP targeting.β Recent research has emphasized the potential importance of the Divisia monetary aggregates in implementing that policy. We investigate bivariate time series properties of Divisia money and nominal GDP to investigate the viability of recent proposals by authors who advocate a role for a Divisia monetary aggregate in nominal GDP targeting.
There are two particularly relevant proposals: (1) the proposal by Barnett, Chauvet, and Leiva-Leon (2015) to use a Divisia monetary aggregate as an indicator in the monthly Nowcasting of nominal GDP, as needed in implementation of any nominal GDP targeting policy; and (2) the proposal by Belongia and Ireland (2015) to use a Divisia monetary aggregate as an intermediate target, with nominal GDP being the final target of policy.
We run well known diagnostic tests of bivariate time series properties of the Divisia M2 and nominal GDP stochastic processes. Those tests are for properties that are necessary, but not sufficient, for the conclusions of Belongia and Ireland (2014) and Barnett, Chauvet, and Leiva-Leon (2015). We find no time series properties that would contradict those implied by either of those two approaches.
Keywords: money, aggregation theory, index number theory, Divisia index, Divisia monetary aggregates, nominal GDP targeting.
JEL Classification Codes: C43, E01, E3, E40, E41, E51, E52, E58.
1. Introduction
The recent financial crisis has induced central banks to explore and
undertake unconventional approaches to monetary policy. One of the hottest topics in monetary policy research has been the revival of the proposal for βnominal GDP targetingβ, advocated by many leading monetary economists, including Michael Woodford, Christina Romer, and Paul Krugman. Proponents argue that nominal GDP targeting can stabilize the macroeconomy more effectively than inflation targeting.
In particular, they argue that by committing to return nominal GDP to its pre-crisis trajectory, the Federal Reserve could improve confidence and expectations of future growth.
We take no position on whether nominal GDP should be adopted as the new monetary policy target, but we investigate the bivariate time series properties of Divisia money and nominal GDP that are relevant to recent results by authors who do advocate a role for a Divisia monetary aggregate in nominal GDP targeting.
There are two such proposals. (1) The least controversial is the approach of
Barnett, Chauvet, and Leiva-Leon (2015) to the use of Divisia money in Nowcasting of nominal GDP. Any approach to targeting nominal GDP requires availability of monthly measurements of nominal GDP. Monthly measurements of nominal GDP are needed regardless of the instrument of policy adopted to implement the targeting. But nominal GDP data are available only quarterly. Using an advanced dynamic factor analysis approach to Nowcasting, Barnett, Chauvet, and Leiva-Leon (2015) find that the most accurate available approach to Nowcasting nominal GDP would use a Divisia monetary aggregate as one of the relevant and highly significant associated variables, with the others being measures of real economic activity and inflation dynamics. While Nowcasting does not imply unidirectional causation, Nowcasting approaches do require existence of strong bivariate time series
associations among the interpolated variable and the associated variables. (2) The more controversial approach, suggesting a monetarist perspective, advocates the
use of a Divisia monetary aggregate as an intermediate target in the procedure for targeting nominal GDP. Such an approach has been advocated by Belongia and Ireland (2015), while a new Keynesian approach has been proposed by the same authors in Belongia and Ireland (2014).
Early suggestions of the possible use of monetary aggregates in nominal GDP targeting include Feldstein and Stock (1993), who showed that the relation between M2 and nominal GDP is sufficiently strong to warrant further investigation into using M2 to influence nominal GDP, as would be relevant to the second approach described above. Since recent research has found Divisia monetary aggregates to be substantially superior to simple sum aggregates, we concentrate in this paper on Divisia M2. See, e.g., Barnett (2012,2015) and Barnett and Chauvet (2011) regarding the superiority of Divisia monetary aggregates over the now largely discredited simple sum monetary aggregates. But since our results are relevant to Nowcasting nominal GDP as well as intermediate targeting, our results are relevant even to proposals in which money is not used to influence nominal GDP, but only to interpolate the quarterly GDP data. In that case, our results need not be interpreted as having implications for the choice of instrument or intermediate targets in the policy rule.
Setting up a VAR model to indicate such relationship, we focus on
ππ(ππππππππππππ) and ππ(ππππππ2), which are the growth rates of nominal GDP and Divisia M2. The estimated model indicates that there is a bidirectional Granger Causality relation between the two. We can make predictions based on our estimated model and can investigate how growth rate of Divisia money supply is going to impact nominal GDP and vice versa. The primary objective of this research is to run well known diagnostic tests of bivariate time series properties of the Divisia M2 and nominal GDP stochastic processes. Those tests are for properties that are necessary, but not sufficient, for the conclusions of Belongia and Ireland (2014) and Barnett, Chauvet, and Leiva-Leon (2015).
2. Literature Review
A nominal GDP target was previously called a βnominal income targetβ by early supporters such as McCallum (2011,2013). This approach is often contrasted with inflation targeting. Under some proposals on nominal GDP targeting, the central bank would try to keep nominal GDP growing at a predetermined rate. A nominal GDP level target is similar, except that the central bank would recall any previous deviations of nominal GDP growth from target and seek to compensate in later years. Apart from Bennett McCallum, who advocates nominal GDP growth rate targeting, most of the current supporters of nominal GDP targeting favor nominal GDP level targeting, such as Woodford (2013), Belongia and Ireland (2015), and Sumner (2012).
Christina Romer (2011), then chair of the Council of Economic Advisers, has urged adopting nominal GDP targeting as the monetary policy rule. In Romerβs view, such a policy would be a powerful communication tool. By pledging to do whatever it takes to return nominal GDP to its pre-crisis trajectory, the Fed could improve confidence and expectations of future growth. Because nominal GDP reflects the Fedβs dual mandate, stable price level and maximum real output, Romer argues that nominal GDP targeting would have a better chance of reducing unemployment than any other monetary policy approach under discussion.
Woodford (2013) argues that long run inflation targeting does not need to be repudiated as a policy framework, but rather needs to be completed. He argues that the target path for nominal GDP could be chosen such that keeping nominal GDP on that path should ensure, over the medium run, an average inflation rate equal to the inflation target. In his view, nominal GDP targeting can complete inflation targeting without conflicting with it. He further maintains that nominal GDP targeting would reduce the tension between the goals of restraining risks to financial stability, on the one hand, and maintaining macroeconomic stability, on the other.2
Sumner (2012), a persistent advocator of nominal GDP targeting and relentless blogger of βThe Money Illusion,β argues that the recent financial crisis
2 Regarding inflation targeting, see, e.g., Bernanke and Mishkin (1997) and Svensson (1998).
exposed serious flaws with inflation targeting monetary policy regimes. In his view, GDP targeting would have greatly reduced the severity of the recession and also eliminated the need for fiscal stimulus. He also argues that nominal GDP targeting would make it easier for politicians to resist calls for bailouts of private sector firms, while assuring low inflation and reducing the severity of the business cycle. He also argues that nominal GDP targeting would make asset price bubbles less likely to occur. In summary, advocates of nominal GDP targeting believe it would provide the best environment for free-market policies to flourish.
On September 12, 2012, the Federal Reserve undertook policy initiatives influenced by Woodford (2003,2005,2012): an open-ended quantitative easing program, in which the amount of purchases depends on progress toward the policy goals. The Federal Reserve also announced it would maintain an easy money policy for some period after the economy has recovered. That announcement can be interpreted as an incremental move toward nominal GDP level targeting.
Nominal GDP targeting defines the final target of policy, but not the instrument, intermediate target, or rule used to implement the final target
commitment. Many proposed approaches exist, including those that implement the final target for a new-Keynesian approach, a post-Keynesian approach, a monetarist approach, a classical approach, a new-classical approach, or an Austrian School approach. McCallum (1987) proposes a monetarist rule that uses the monetary base as instrument to target nominal GDP. He advocates targeting the growth rate of nominal GDP, rather than the level. His view is that if growth rates are on average equal to the target value over time, the policy would be unlikely to permit much departure from the planned path and should therefore be preferred. His rule employs a four-year moving average of past growth in monetary base velocity to forecast that velocityβs growth in the coming quarter. Based on that forecast, the rule specifies the percentage of the gap between the targeted and actual levels of nominal GDP that the central bank should plan to close in the coming quarter.
In simulations, Dueker (1993) confronts McCallumβs nominal GDP targeting rule with a world in which coefficients in the velocity equation for the monetary
instrument are subject to unpredictable stochastic change. His approach differs from McCallumβs by using explanatory variables to help forecast velocity in a time- varying parameter model. By allowing for time-varying coefficients, Duekerβs forecasting model is argued to be more stable over time than fixed-coefficient models. Dueker concludes that McCallumβs approach to nominal GDP targeting is simple yet robust to velocity behavior. However, Duekerβs forecast-based rule performed somewhat better in simulations in which velocity was generated from a time-varying parameter model.
Recent contributors to the literature on nominal GDP targeting also incorporate aggregation theoretic monetary aggregates. Belongia and Ireland (2015) derive an approach to targeting the level of nominal GDP using a framework first outlined by Working (1923) and used, with minor modifications, by Hallman, et al. (1991) in their P-Star model. Belongia and Irelandβs framework is built on
traditional quantity theoretic foundations and draws directly from Barnettβs (1978,1980) economic approach to monetary aggregation. With any desired long- run trajectory for nominal GDP, the framework can find a consistent intermediate target path for Divisia money. The central bank can use the monetary base to control the intermediate target path for either a narrow or broad Divisia monetary aggregate and thereby keep nominal GDP growing along any desired long-run path.
Their innovation lies in employing Divisia monetary aggregates to establish a path for the intermediate target and uses a one-sided filtering algorithm to control for slow-moving trends in velocity. The merits of this approach are its transparency to outside observers, its forward-looking design, and its potentially straightforward implementation.
Barnett, Chauvet, and Leiva-Leon (2015) developed dynamic factor models to Nowcast nominal output growth, using information from the previous release of nominal GDP, Industrial Production, Consumer Price Index, and Divisia M3. Their model is useful in giving monthly assessment of the current nominal GDP quarterly growth. This ability plays an essential role in monitoring the effectiveness of nominal GDP targeting monetary policy, regardless of the approach to
implementation. In fact any approach that uses monthly feedback in its nominal GDP targeting approach becomes undefined, and thereby not applicable, without access to monthly GDP Nowcasts.
3. The Bivariate Time Series Relationship between Divisia M2 and Nominal GDP
As explained above, the use of Divisia monetary aggregates has been
proposed in two different potential roles in nominal GDP targeting. One role is as an indicator variable in Nowcasting of monthly nominal GDP, as needed in any
implementation of nominal GDP targeting. The other roles is direct use as an intermediate target in the policy design. Both cases imply the existence of a bivariate time series relationship between a Divisia monetary aggregate and nominal GDP. In this paper, we explore the nature of that relationship.
The Divisia monetary aggregate we use is Divisia M2, as provided by the Federal Reserve Bank of St. Louis in its FRED database. We use those data since they are well known and have a long history in this literature. But in future research, we plan to use the broader Divisia monetary aggregates, M3 and M4, supplied by the Center for Financial Stability in New York City.3 The GDP data we use are supplied by the U.S. Bureau of Economic Analysis (BEA). Both series are seasonally adjusted. We eliminate heteroskedasticity by taking logarithms of the variables. We use ππππππππππππ and ππππππ2 to denote the transformed data.
3.1. Unit Root Test
First we conduct a unit root test to examine stationarity of the series. If the series are non-stationary, regression could be spurious. We adopt the ADF
(Augmented Dickey-Fuller) method for unit root test. The test results are displayed in the appendix as Table 1a.
3 See Barnett, Liu, Mattson, and van den Noort (2013).
The p values of both tests are greater than the 5% significance level, with 0.9951 for ππππππππππππ and 0.4876 for ππππππ2 respectively. Hence, for each of the tests, we fail to reject the null hypothesis that the series has a unit root. Both ππππππππππππ and ππππππ2 series are non-stationary.
To test for causality relationship between nominal GDP and Divisia M2 money supply, we need the series to be stationary. For that purpose, we first difference the series to produce two first order differenced series ππ(ππππππππππππ) and ππ(ππππππ2). We then again conduct the ADF test on each of those transformed series.
The null hypotheses that ππ(ππππππππππππ) and ππ(ππππππ2) have unit roots are decisively rejected. The differenced time series are stationary processes. See Table 2a in the appendix.
3.2. Cointegration Test
Next we test cointegration between ππππππππππππ and ππππππ2 to investigate whether there exists long run association between the two processes. If the two variables are not cointegrated, we could apply an unrestricted VAR model. If the variables are cointegrated, we should prefer a vector error correction model (VECM). We use Johansenβs (1988,1991) methodology. The p values for unrestricted cointegration rank tests using trace and maximum eigenvalue are 0.0828 and 0.0646 respectively, both higher than 5% significance level. See Table 3a in the appendix. Hence we fail to reject the null hypothesis of no cointegration between ππππππππππππ and ππππππ2. We use an unrestricted VAR model in the following step.
3.3. VAR Model
We begin with a preliminary unrestricted VAR(2) model, as shown in appendix table 4a. We use the Akaike Information Criterion (AIC) to determine the appropriate maximum lag length for the variables in the VAR. Since we are using quarterly data, we choose lag equal to 4, when conducting VAR lag order selection.
As the following table 1 shows, lag equal to 3 gives us the lowest AIC value.
Therefore, we revise our model to a VAR(3) and estimate its coefficients. Detailed results are in appendix table 5a.
Table 1: VAR Lag Order Selection Criteria
Endogenous variables: ππ(ππππππ2), ππ(ππππππππππππ) Sample: 1967Q1 - 2013Q4
Included observations: 183
Lag Log L LR FPE AIC SC HQ
0 1201.558 NA 6.94e-09 -13.10992 -13.07485 -13.09571
1 1259.317 113.6242 3.86e-09 -13.69745 -13.59222 -13.65480
2 1269.885 20.55929* 3.59e-09 -13.76924 -13.59386* -13.69815*
3 1274.130 8.164230 3.58e-09* -13.77191* -13.52638 -13.67238
4 1277.989 7.338699 3.59e-09 -13.77037 -13.45468 -13.64241
* Identifies the lag order selected by the criterion in that column.
Log L: log likelihood
LR: sequential modified LR test statistic (each test at 5% level) FPE: Final prediction error
AIC: Akaike information criterion SC: Schwarz information criterion HQ: Hannan-Quinn information criterion
Next we examine whether there exist autocorrelation problems among the disturbances. Using the Autocorrelation LM (Lagrange Multiplier) test with lag equal to 12, we acquire the following table 2 with most of the p values greater than the 5%
significance level.
Table 2: VAR Residual Serial Correlation LM Tests
Null Hypothesis: no serial correlation at lag order
Sample: 1967Q1 - 2013Q4
Included observations: 184
Lags LM-Statistic P value*
1 8.170979 0.0855
2 10.45168 0.0335
3 6.668278 0.1545
4 6.192919 0.1852
5 10.20056 0.0372
6 7.367825 0.1177
7 2.768448 0.5973
8 4.482638 0.3446
9 9.023472 0.0605
10 1.994479 0.7368
11 12.65099 0.0131
12 5.147886 0.2725
*P value from chi-square with 4 degrees of
freedom.
We fail to reject the null hypothesis of no serial correlation among the residuals of the VAR(3) model. The VAR(3) model is well-specified.
3.4. Granger Causality Test
We conducted Granger causality tests between ππ(ππππππππππππ) and ππ(ππππππ2).
The results indicate that ππ(ππππππππππππ) Granger causes ππ(ππππππ2), and ππ(ππππππ2) also Granger Causes ππ(ππππππππππππ). Listed below in table 3 are the Granger causality test results.
Table 3: VAR Granger Causality, Block Exogeneity Wald Tests
Sample: 1967Q1 2013Q4 Included observations: 184
Dependent variable: ππ(ππππππ2)
Excluded Chi-sq df P value
ππ(ππππππππππππ) 11.28757 3 0.0103
All 11.28757 3 0.0103
Dependent variable: ππ(ππππππππππππ)
Excluded Chi-sq df P value
ππ(ππππππ2) 11.67938 3 0.0086
All 11.67938 3 0.0086
The P value of the null hypothesis that ππ(ππππππππππππ) does not Granger cause ππ(ππππππ2) is 0.0103, which is smaller than the conventional critical value 0.05. We reject the null and therefore conclude that ππ(ππππππππππππ) does Granger cause ππ(ππππππ2). The P value of the null hypothesis that ππ(ππππππ2) does not Granger cause ππ(ππππππππππππ) is 0.0086, also smaller than the critical value 0.05. We reject the null hypothesis and therefore conclude that ππ(ππππππ2) does Granger cause ππ(ππππππππππππ).
There exists a bidirectional Granger causality relationship between ππ(ππππππππππππ) and ππ(ππππππ2).
3.5. Estimation of the Final Bivariate VAR
We implemented the bidirectional Granger Causality relationship between ππ(ππππππππππππ) and ππ(ππππππ2) by estimating a bivariate VAR in those two stochastic processes with optimized lag lengths selected from the EViews program. The coefficients of the two equations are stacked into one vector having elements, C(i), i
= 1, β¦ , 14, as defined in table 6a in the appendix. The two equations we estimated in this VAR are defined in Table 6a. The coefficients of the first equation are C(i), i = 1, β¦ , 7 , while the coefficients of the second equation are C(i), i = 8, β¦ , 14. See the table for the specification of those two equations and the estimates of their
coefficients.
The p value for πΆπΆ(1) is 0.0000, demonstrating that the coefficient of
ππ(ππππππ2)π‘π‘β1 is significant in the first equation. The growth rate of Divisa M2 money supply in the previous period has a significant impact on prediction of the current growth rate of Divisia M2. The corresponding p value of πΆπΆ(2) is 0.9735,
demonstrating that the second lag of the growth rate of M2 does not have significant predicting power for the current growth rate of M2. By eliminating the statistically insignificant coefficients, we acquire the following two estimated equations:
ππ(ππππππ2)π‘π‘ = 0.483728ππ(ππππππ2)π‘π‘β1+ 0.146457ππ(ππππππ2)π‘π‘β3β
β 0.223671ππ(ππππππππππππ)π‘π‘β1+ 0.006672 (1)
ππ(ππππππππππππ)π‘π‘= 0.223336ππ(ππππππ2)π‘π‘β1+ 0.318158ππ(ππππππππππππ)π‘π‘β1
+0.288470ππ(ππππππππππππ)π‘π‘β2 . (2)
Since ππ(ππππππ2) and ππ(ππππππππππππ) indicate the growth rates, the estimated equations can be interpreted as follows. The growth rate of Divisia M2 is affected by the growth rate of itself, lagged by 1 and 3 quarters, as well as by the growth rate of the previous quarterβs nominal GDP. Furthermore, holding other variables constant, we can reach the following conclusions. From the first equation, if the growth rate of Divisia M2 during the last quarter increases by 10%, then the growth rate of M2 this quarter will increase by 4.83728%. But if the nominal GDP growth rate of the previous quarter increases by 10%, the M2 growth rate this quarter will decrease by 2.23671%. If the M2 growth rate, lagged three quarters, reaches 10%, the current growth rate will increase by 1.46457%. Similar analysis applies to the second equation, where ππ(ππππππππππππ)π‘π‘ is the dependent variable.
3.6. Prediction
Based on the estimation of equations (1) and (2), we can predict the growth rate of Divisia M2 and nominal GDP in 2014 Q1 using the quarterly data in our sample ending in 2013 Q4.4
β©βͺ
β¨
βͺβ§ ππ(ππππππ2)2014ππ1 = 0.483728ππ(ππππππ2)2013ππ4+ 0.146457ππ(ππππππ2)2013ππ2β 0.223671ππ(ππππππππππππ)2013ππ4+ 0.006672
ππ(ππππππππππππ)2014ππ1= 0.223336ππ(ππππππ2)2013ππ4+ 0.318158ππ(ππππππππππππ)2013ππ4 +0.288470ππ(ππππππππππππ)2013ππ2
Substituting the measured values of the variables into the right hand sides, the predicted growth rates are:
οΏ½ ππ(ππππππ2)2014ππ1= 0.014079 ππ(ππππππππππππ)2014ππ1 = 0.012193
The predicted growth rates can be used to predict the levels of M2 and NGDP in 2014Q1 by the following equations:
οΏ½ ππ22014ππ1= ππ22013ππ4β(1 +ππ(ππππππ2)2014ππ1) ππππππππ2014ππ1 =ππππππππ2013ππ4β(1 +ππ(ππππππππππππ)2014ππ1) Substituting into the right hand sides, we acquire:
οΏ½ ππ22014ππ1= 11758.8 ππππππππ2014ππ1 = 17286.5
The 1.4% predicted growth rate of Divisia M2 money supply in 2014Q1 was inconsistence with the Federal Reserveβs accommodative monetary policy. A consequence is reflected in the almost-non-growing 1.2% nominal GDP prediction in 2014Q1. In fact, the out of sample growth rate of 2014Q1 was -0.2%, according to the data released by Bureau of Economic Analysis (BEA).
4 We could have used a longer sample period including more recent quarters by using data from the Center for Financial Stability (CFS) in New York City. But we limited this study to data made available by the Federal Reserve Bank of St. Louis, which has not updated its data as regularly as the CFS, which does update monthly.
4. Conclusion
In this paper we discuss the relationship between Divisia M2 money supply and nominal GDP. The primary objective of this research is to run well known diagnostic tests of bivariate time series properties of the Divisia M2 and nominal GDP stochastic processes that are necessary but not sufficient for the conclusions of Belongia and Ireland (2014) and Barnett, Chauvet, and Leiva-Leon (2015). We find no evidence to contradict the conclusions of those two papers about the potential relevancy of Divisia monetary aggregates in targeting nominal GDP, either as an intermediate target or as an indicator. Our results are not specific to either of those approaches and hence cannot provide conclusions about which of those two
approaches should be preferred. Since neither of those two approaches contradicts the other, one possibility would be to use both of those approaches simultaneously.
In that case, Barnett, Chauvet, and Leiva-Leon (2015) could be used to interpolate the quarterly data to provide the needed Nowcast monthly nominal GDP data, while Belongia and Ireland (2014) would then be used to implement a policy design using a Divisia monetary aggregate as an intermediate target.
But if a different policy design were adopted without an intermediate target, Barnett, Chauvet, and Leiva-Leon (2015) would remain relevant to producing the monthly data necessary for any approach to nominal GDP targeting.
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Appendix
Table 1a. Unit Root Test Result for ππππππππππππ and ππππππ2
Null Hypothesis: ππππππππππππhas a unit root
t-Statistic P value*
Augmented Dickey-Fuller test statistic -0.065053 0.9951
Test critical values: 1% level -4.008154
5% level -3.434167
10% level -3.141001
*MacKinnon (1996) one-sided p-values.
Null Hypothesis: ππππππ2has a unit root
t-Statistic P value*
Augmented Dickey-Fuller test statistic -2.197872 0.4876
Test critical values: 1% level -4.008154
5% level -3.434167
10% level -3.141001
*MacKinnon (1996) one-sided p-values.
Table 2a. Unit Root Test Result for ππ(ππππππππππππ) and ππ(ππππππ2)
Null Hypothesis: ππ(ππππππππππππ) has a unit root
t-Statistic P value*
Augmented Dickey-Fuller test statistic -10.34110 0.0000
Test critical values: 1% level -4.008154
5% level -3.434167
10% level -3.141001
*MacKinnon (1996) one-sided p-values.
Null Hypothesis: ππ(ππππππ2) has a unit root
t-Statistic P value*
Augmented Dickey-Fuller test statistic -7.718251 0.0000
Test critical values: 1% level -4.008154
5% level -3.434167
10% level -3.141001
*MacKinnon (1996) one-sided p-values.
Table 3a. Johansen Cointegration Test Between ππππππππππππ and ππππππ2
Sample (adjusted): 1968Q2 - 2013Q4 Included observations: 183
Series: ππ(ππππππ2), ππ(ππππππππππππ)
Lags interval (in first differences): 1 to 4 Unrestricted Cointegration Rank Test (Trace)
Hypothesized Trace 0.05
No. of CE(s) Eigenvalue Statistic Critical Value P value**
None 0.071362 14.00454 15.49471 0.0828
At most 1 0.002488 0.455880 3.841466 0.4996
Trace test indicates no cointegration at the 0.05 level *Denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values
Unrestricted Cointegration Rank Test (Maximum Eigenvalue)
Hypothesized Max-Eigen 0.05
No. of CE(s) Eigenvalue Statistic Critical Value P value**
None 0.071362 13.54866 14.26460 0.0646
At most 1 0.002488 0.455880 3.841466 0.4996
Max-eigenvalue test indicates no cointegration at the 0.05 level *Denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values Unrestricted Cointegrating Coefficients:
ππππππ2 ππππππππππππ
1.852061 -3.068247 9.671655 -7.514269
Unrestricted Adjustment Coefficients (alpha):
ππ(ππππππ2) 0.000987 0.000297
ππ(ππππππππππππ) 0.001715 -0.000217
One Cointegrating Equation: Log likelihood 1284.763 Normalized cointegrating coefficients (standard error in parentheses)
ππππππ2 ππππππππππππ
1.000000 -1.656666 (0.23697)
Adjustment coefficients (standard error in parentheses) ππ(ππππππ2) 0.001828
(0.00098) ππ(ππππππππππππ) 0.003177 (0.00106)
Table 4a. VAR(2) Estimation
Vector Autoregression Estimates Sample (adjusted): 1967Q4 - 2013Q4 Included observations: 185
Standard errors in ( ) & t-statistics in [ ]
ππ(ππππππππππππ) ππ(ππππππ2)
ππ(ππππππππππππ)π‘π‘β1 0.343726 -0.203535
(0.07111) (0.06393) [ 4.83377] [-3.18391]
ππ(ππππππππππππ)π‘π‘β2 0.296876 0.110804
(0.07117) (0.06398) [ 4.17157] [ 1.73192]
ππ(ππππππ2)π‘π‘β1 0.230714 0.502884
(0.08198) (0.07370) [ 2.81432] [ 6.82361]
ππ(ππππππ2)π‘π‘β2 -0.018083 0.051093
(0.08169) (0.07344) [-0.22137] [ 0.69573]
Constant intercept 0.002710 0.007922 (0.00177) (0.00159) [ 1.53297] [ 4.98427]
R-squared 0.320419 0.300597
Adj. R-squared 0.305317 0.285055
Sum sq. residuals 0.011651 0.009416
S.E. equation 0.008045 0.007233
F-statistic 21.21725 19.34060
Log likelihood 632.2211 651.9214
Akaike AIC -6.780769 -6.993745
Schwarz SC -6.693732 -6.906708
Mean dependent 0.016113 0.014431
S.D. dependent 0.009653 0.008554
Determinant residual covariance (df adj) 3.38E-09 Determinant residual covariance 3.20E-09
Log likelihood 1284.335
Akaike information criterion -13.77659
Schwarz criterion -13.60252
Table 5a. VAR(3) Estimation
Sample (adjusted): 1968Q1 - 2013Q4 Included observations: 184
Standard errors in ( ) and t-statistics in [ ]
ππ(ππππππππππππ) ππ(ππππππ2)
ππ(ππππππππππππ)π‘π‘β1 0.318158 -0.223671
(0.07460) (0.06667) [ 4.26460] [-3.35472]
ππ(ππππππππππππ)π‘π‘β2 0.288470 0.062865
(0.07726) (0.06904) [ 3.73398] [ 0.91053]
ππ(ππππππππππππ)π‘π‘β3 0.076208 0.074424
(0.07535) (0.06734) [ 1.01134] [ 1.10515]
ππ(ππππππ2)π‘π‘β1 0.223336 0.483728
(0.08300) (0.07418) [ 2.69084] [ 6.52140]
ππ(ππππππ2)π‘π‘β2 0.061580 0.002791
(0.09397) (0.08398) [ 0.65531] [ 0.03323]
ππ(ππππππ2)π‘π‘β3 -0.113475 0.146457
(0.08194) (0.07323) [-1.38480] [ 1.99990]
Constant intercept 0.002610 0.006672 (0.00190) (0.00170) [ 1.37251] [ 3.92561]
R-squared 0.331774 0.320114
Adj. R-squared 0.309123 0.297067
Sum sq. residuals 0.011451 0.009146
S.E. equation 0.008043 0.007188
F-statistic 14.64676 13.88960
Log likelihood 629.8995 650.5791
Akaike AIC -6.770647 -6.995425
Schwarz SC -6.648340 -6.873118
Mean dependent 0.016098 0.014412
S.D. dependent 0.009677 0.008574
Determinant residual covariance (df adj.) 3.34E-09 Determinant residual covariance 3.09E-09
Log likelihood 1280.607
Akaike information criterion -13.76747
Schwarz criterion -13.52286
Table 6a. Final VAR Coefficient Estimation
Estimation Software: EViews computer program Sample: 1968Q1 - 2013Q4
Included observations: 184 Total system observations 368
Coefficient Std. Error t Statistic P Value
C(1) 0.483728 0.074176 6.521397 0.0000
C(2) 0.002791 0.083982 0.033235 0.9735
C(3) 0.146457 0.073232 1.999902 0.0463
C(4) -0.223671 0.066674 -3.354718 0.0009
C(5) 0.062865 0.069043 0.910526 0.3632
C(6) 0.074424 0.067343 1.105148 0.2698
C(7) 0.006672 0.001700 3.925608 0.0001
C(8) 0.223336 0.082999 2.690835 0.0075
C(9) 0.061580 0.093971 0.655308 0.5127
C(10) -0.113475 0.081943 -1.384797 0.1670
C(11) 0.318158 0.074604 4.264599 0.0000
C(12) 0.288470 0.077255 3.733981 0.0002
C(13) 0.076208 0.075353 1.011342 0.3125
C(14) 0.002610 0.001902 1.372509 0.1708
Equation: ππ(ππππππ2)π‘π‘ = C(1)ππ(ππππππ2)π‘π‘β1 + C(2)ππ(ππππππ2)π‘π‘β2 + C(3)ππ(ππππππ2)π‘π‘β3 + C(4)ππ(ππππππππππππ)π‘π‘β1+ C(5)ππ(ππππππππππππ)π‘π‘β2 + C(6)ππ(ππππππππππππ)π‘π‘β3 + C(7)
R squared 0.320114 Mean dependent var 0.014412
Adjusted R squared 0.297067 dependent var 0.008574
S.E. of regression 0.007188 Sum squared resid 0.009146 Durbin-Watson stat 1.977980
Equation: ππ(ππππππππππππ)π‘π‘= C(8)ππ(ππππππ2)π‘π‘β1 + C(9)ππ(ππππππ2)π‘π‘β2 + C(10)ππ(ππππππ2)π‘π‘β3+ C(11)ππ(ππππππππππππ)π‘π‘β1 + C(12)ππ(ππππππππππππ)π‘π‘β2 + C(13)ππ(ππππππππππππ)π‘π‘β3 + C(14)
R squared 0.331774 Mean dependent var 0.016098
Adjusted R-squared 0.309123 S.D. dependent var 0.009677 S.E. of regression 0.008043 Sum squared resid 0.011451 Durbin Watson stat 1.998140